Solve the following question

In a study conducted by the Department of Mechanical Engineering at Virginia Tech, the steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company, and a measure of flexibility was recorded for each. The data are as follows: Company A: 9.3 8.8 6.8 8.7 8.5 6.7 8.0 6.5 9.2 7.0 Company B: 11.0 9.8 9.9 10.2 10.1 9.7 11.0 11.1 10.2 9.6 (a) Calculate the sample mean and median for the data for the two companies. (b) Plot the data for the two companies on the same line and give your impression regarding any apparent differences between the two companies. An overseas shipment of 5 foreign automobiles contains 2 that have slight paint blemishes. If an agency receives 3 of these automobiles at random, list the elements of the sample space S, using the letters Band / for blemished and nonblemished, respectively; then to each sample point assign a value x of the random variable X representing the number of automobiles with paint blemishes purchased by the agency.Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error, and statisticians spend a great deal of time modeling these errors. Suppose the measurement error X of a certain physical quantity is decided by the density function f(x) = [*(3-r'), -15251, elsewhere. (a) Determine * that renders f(x) a valid density function.(b) Find the probability that a random error in measurement is less than 1/2.(c) For this particular measurement, it is undesirable if the magnitude of the error (ie., (x]) exceeds 0.8. What is the probability that this occurs? Five independent coin tosses result in HHHHH. It turns out that if the coin is fair the probability of this outcome is (1/2)5 = 0.03125. Does this produce strong evidence that the coin is not fair? Comment and use the concept of P-value discussed inSection 1.1.From a box containing 4 dimes and 2 nickels, 3 coins are selected at random without replacement. Find the probability distribution for the total 7 of the 3 coins. Express the probability distribution graphically as a probability histogram. Consider the birthdays of the students in a class of size r. Assume that the year consists of 365 days. (a) How many different ordered samples of birthdays are possible (r in sample) allowing repetitions (with replacement)? (b) The same as part (a), except requiring that all the students have different birthdays (without replacement)? (c) If we can assume that each ordered outcome in part (a) has the same probability, what is the probability that at least two students have the same birthday? (d) For what value of r is the probability in part (c) about equal to 1/2? Is this number surprisingly small? Hint: Use a calculator or computer to find r.Find the probability distribution for the number of jazz CDs when 4 CDs are selected at random from a collection consisting of 5 jazz CDs, 2 classical CDs, and 3 rock CDs. Express your results by means of a formula. A tire manufacturer wants to determine the inner diameter of a certain grade of tire. Ideally, the diameter would be 570 mm. The data are as follows: 572, 572, 573, 568, 569, 575, 565, 570. (a) Find the sample mean and median. (b) Find the sample variance, standard deviation, and range. (c) Using the calculated statistics in parts (a) and (b), can you comment on the quality of the tires?The proportion of the budget for a certain type of industrial company that is allotted to environmental and pollution control is coming under scrutiny. A data collection project determines that the distribution of these proportions is given by f(y) = 5(1-V)'. OSys1. elsewhere. (a) Verify that the above is a valid density function. (b) What is the probability that a company chosen at random expends less than 10% of its budget on environmental and pollution controls? (c) What is the probability that a company selected at random spends more than 50% of its budget on environmental and pollution controls